{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T19:39:44Z","timestamp":1765827584528},"reference-count":46,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>An optimally convergent (with respect to the regularity) quadratic\nfinite element method for the two-dimensional obstacle problem on\nsimplicial meshes is studied in [14]. There was no analogue of a quadratic\nfinite element method on tetrahedron meshes for the three-dimensional\nobstacle problem. In this article, a quadratic finite element\nenriched with element-wise bubble functions is proposed for the\nthree-dimensional elliptic obstacle problem. A priori error\nestimates are derived to show the optimal convergence of the\nmethod with respect to the regularity. Further, a posteriori error\nestimates are derived to design an adaptive mesh refinement\nalgorithm. A numerical experiment illustrating the theoretical\nresult on a priori error estimates is presented.<\/jats:p>","DOI":"10.1515\/cmam-2017-0018","type":"journal-article","created":{"date-parts":[[2017,7,13]],"date-time":"2017-07-13T10:02:24Z","timestamp":1499940144000},"page":"223-236","source":"Crossref","is-referenced-by-count":12,"title":["Bubbles Enriched Quadratic Finite Element Method for the 3D-Elliptic Obstacle Problem"],"prefix":"10.1515","volume":"18","author":[{"given":"Sharat","family":"Gaddam","sequence":"first","affiliation":[{"name":"Department of Mathematics , Indian Institute of Science , Bangalore 560012 , India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Thirupathi","family":"Gudi","sequence":"additional","affiliation":[{"name":"Department of Mathematics , Indian Institute of Science , Bangalore 560012 , India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,7,11]]},"reference":[{"key":"2023033109580933521_j_cmam-2017-0018_ref_001_w2aab3b7d994b1b6b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"M.  Ainsworth and J. T.  Oden,\nA Posteriori Error Estimation in Finite Element Analysis,\nPure Appl. Math. (New York),\nWiley-Interscience, New York, 2000.","DOI":"10.1002\/9781118032824"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_002_w2aab3b7d994b1b6b1ab2ab2Aa","doi-asserted-by":"crossref","unstructured":"M.  Ainsworth, J. T.  Oden and C.-Y.  Lee,\nLocal a posteriori error estimators for variational inequalities,\nNumer. Methods Partial Differential Equations 9 (1993), no. 1, 23\u201333.","DOI":"10.1002\/num.1690090104"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_003_w2aab3b7d994b1b6b1ab2ab3Aa","unstructured":"K.  Atkinson and W.  Han,\nTheoretical Numerical Analysis, 3rd ed.,\nTexts Appl. Math. 39,\nSpringer, New York, 2009."},{"key":"2023033109580933521_j_cmam-2017-0018_ref_004_w2aab3b7d994b1b6b1ab2ab4Aa","doi-asserted-by":"crossref","unstructured":"L.  Banz and E. P.  Stephan,\nA posteriori error estimates of hp-adaptive IPDG-FEM for elliptic obstacle problems,\nAppl. Numer. Math. 76 (2014), 76\u201392.","DOI":"10.1016\/j.apnum.2013.10.004"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_005_w2aab3b7d994b1b6b1ab2ab5Aa","doi-asserted-by":"crossref","unstructured":"S.  Bartels and C.  Carstensen,\nAveraging techniques yield reliable a posteriori finite element error control for obstacle problems,\nNumer. Math. 99 (2004), no. 2, 225\u2013249.","DOI":"10.1007\/s00211-004-0553-6"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_006_w2aab3b7d994b1b6b1ab2ab6Aa","doi-asserted-by":"crossref","unstructured":"F.  Ben Belgacem,\nNumerical simulation of some variational inequalities arisen from unilateral contact problems by the finite element methods,\nSIAM J. Numer. Anal. 37 (2000), no. 4, 1198\u20131216.","DOI":"10.1137\/S0036142998347966"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_007_w2aab3b7d994b1b6b1ab2ab7Aa","doi-asserted-by":"crossref","unstructured":"H.  Blum and F.-T.  Suttmeier,\nAn adaptive finite element discretisation for a simplified Signorini problem,\nCalcolo 37 (2000), no. 2, 65\u201377.","DOI":"10.1007\/s100920070008"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_008_w2aab3b7d994b1b6b1ab2ab8Aa","doi-asserted-by":"crossref","unstructured":"D.  Braess,\nA posteriori error estimators for obstacle problems\u2014Another look,\nNumer. Math. 101 (2005), no. 3, 415\u2013421.","DOI":"10.1007\/s00211-005-0634-1"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_009_w2aab3b7d994b1b6b1ab2ab9Aa","doi-asserted-by":"crossref","unstructured":"D.  Braess, C.  Carstensen and R. H. W.  Hoppe,\nConvergence analysis of a conforming adaptive finite element method for an obstacle problem,\nNumer. Math. 107 (2007), no. 3, 455\u2013471.","DOI":"10.1007\/s00211-007-0098-6"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_010_w2aab3b7d994b1b6b1ab2ac10Aa","unstructured":"D.  Braess, C.  Carstensen and R. H. W.  Hoppe,\nError reduction in adaptive finite element approximations of elliptic obstacle problems,\nJ. Comput. Math. 27 (2009), no. 2\u20133, 148\u2013169."},{"key":"2023033109580933521_j_cmam-2017-0018_ref_011_w2aab3b7d994b1b6b1ab2ac11Aa","doi-asserted-by":"crossref","unstructured":"S. C.  Brenner and L. R.  Scott,\nThe Mathematical Theory of Finite Element Methods, 3rd ed.,\nTexts Appl. Math. 15,\nSpringer, New York, 2008.","DOI":"10.1007\/978-0-387-75934-0"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_012_w2aab3b7d994b1b6b1ab2ac12Aa","doi-asserted-by":"crossref","unstructured":"S. C.  Brenner, L.  Sung and Y.  Zhang,\nFinite element methods for the displacement obstacle problem of clamped plates,\nMath. Comp. 81 (2012), no. 279, 1247\u20131262.","DOI":"10.1090\/S0025-5718-2012-02602-0"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_013_w2aab3b7d994b1b6b1ab2ac13Aa","doi-asserted-by":"crossref","unstructured":"S. C.  Brenner, L.-Y.  Sung, H.  Zhang and Y.  Zhang,\nA quadratic C0C^{0} interior penalty method for the displacement obstacle problem of clamped Kirchhoff plates,\nSIAM J. Numer. Anal. 50 (2012), no. 6, 3329\u20133350.","DOI":"10.1137\/110845926"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_014_w2aab3b7d994b1b6b1ab2ac14Aa","doi-asserted-by":"crossref","unstructured":"F.  Brezzi, W. W.  Hager and P.-A.  Raviart,\nError estimates for the finite element solution of variational inequalities,\nNumer. Math. 28 (1977), no. 4, 431\u2013443.","DOI":"10.1007\/BF01404345"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_015_w2aab3b7d994b1b6b1ab2ac15Aa","doi-asserted-by":"crossref","unstructured":"Z.  Chen and R. H.  Nochetto,\nResidual type a posteriori error estimates for elliptic obstacle problems,\nNumer. Math. 84 (2000), no. 4, 527\u2013548.","DOI":"10.1007\/s002110050009"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_016_w2aab3b7d994b1b6b1ab2ac16Aa","doi-asserted-by":"crossref","unstructured":"P. G.  Ciarlet,\nThe Finite Element Method for Elliptic Problems,\nNorth-Holland Publishing, Amsterdam, 1978.","DOI":"10.1115\/1.3424474"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_017_w2aab3b7d994b1b6b1ab2ac17Aa","doi-asserted-by":"crossref","unstructured":"G.  Drouet and P.  Hild,\nOptimal convergence for discrete variational inequalities modelling Signorini contact in 2D and 3D without additional assumptions on the unknown contact set,\nSIAM J. Numer. Anal. 53 (2015), no. 3, 1488\u20131507.","DOI":"10.1137\/140980697"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_018_w2aab3b7d994b1b6b1ab2ac18Aa","doi-asserted-by":"crossref","unstructured":"R. S.  Falk,\nError estimates for the approximation of a class of variational inequalities,\nMath. Comput. 28 (1974), 963\u2013971.","DOI":"10.1090\/S0025-5718-1974-0391502-8"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_019_w2aab3b7d994b1b6b1ab2ac19Aa","doi-asserted-by":"crossref","unstructured":"M.  Feischl, M.  Page and D.  Praetorius,\nConvergence of adaptive FEM for some elliptic obstacle problem with inhomogeneous Dirichlet data,\nInt. J. Numer. Anal. Model. 11 (2014), no. 1, 229\u2013253.","DOI":"10.1002\/pamm.201110373"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_020_w2aab3b7d994b1b6b1ab2ac20Aa","doi-asserted-by":"crossref","unstructured":"R.  Glowinski,\nNumerical Methods for Nonlinear Variational Problems. Reprint of the 1984 original,\nSci. Comput.,\nSpringer, Berlin, 2008.","DOI":"10.1007\/978-3-662-12613-4"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_021_w2aab3b7d994b1b6b1ab2ac21Aa","doi-asserted-by":"crossref","unstructured":"T.  Gudi and K.  Porwal,\nA posteriori error control of discontinuous Galerkin methods for elliptic obstacle problems,\nMath. Comp. 83 (2014), no. 286, 579\u2013602.","DOI":"10.1090\/S0025-5718-2013-02728-7"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_022_w2aab3b7d994b1b6b1ab2ac22Aa","doi-asserted-by":"crossref","unstructured":"T.  Gudi and K.  Porwal,\nA remark on the a posteriori error analysis of discontinuous Galerkin methods for the obstacle problem,\nComput. Methods Appl. Math. 14 (2014), no. 1, 71\u201387.","DOI":"10.1515\/cmam-2013-0015"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_023_w2aab3b7d994b1b6b1ab2ac23Aa","doi-asserted-by":"crossref","unstructured":"T.  Gudi and K.  Porwal,\nA reliable residual based a posteriori error estimator for a quadratic finite element method for the elliptic obstacle problem,\nComput. Methods Appl. Math. 15 (2015), no. 2, 145\u2013160.","DOI":"10.1515\/cmam-2015-0005"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_024_w2aab3b7d994b1b6b1ab2ac24Aa","doi-asserted-by":"crossref","unstructured":"J.  Gwinner,\nOn the p-version approximation in the boundary element method for a variational inequality of the second kind modelling unilateral contact and given friction,\nAppl. Numer. Math. 59 (2009), no. 11, 2774\u20132784.","DOI":"10.1016\/j.apnum.2008.12.027"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_025_w2aab3b7d994b1b6b1ab2ac25Aa","doi-asserted-by":"crossref","unstructured":"J.  Gwinner,\nhp-FEM convergence for unilateral contact problems with Tresca friction in plane linear elastostatics,\nJ. Comput. Appl. Math. 254 (2013), 175\u2013184.","DOI":"10.1016\/j.cam.2013.03.013"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_026_w2aab3b7d994b1b6b1ab2ac26Aa","doi-asserted-by":"crossref","unstructured":"P.  Hild and S.  Nicaise,\nA posteriori error estimations of residual type for Signorini\u2019s problem,\nNumer. Math. 101 (2005), no. 3, 523\u2013549.","DOI":"10.1007\/s00211-005-0630-5"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_027_w2aab3b7d994b1b6b1ab2ac27Aa","doi-asserted-by":"crossref","unstructured":"P.  Hild and Y.  Renard,\nAn improved a priori error analysis for finite element approximations of Signorini\u2019s problem,\nSIAM J. Numer. Anal. 50 (2012), no. 5, 2400\u20132419.","DOI":"10.1137\/110857593"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_028_w2aab3b7d994b1b6b1ab2ac28Aa","doi-asserted-by":"crossref","unstructured":"M.  Hinterm\u00fcller, K.  Ito and K.  Kunish,\nThe primal-dual active set strategy as a semismooth Newton method,\nSIAM J. Optim. 13 (2003), 865\u2013888.","DOI":"10.1137\/S1052623401383558"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_029_w2aab3b7d994b1b6b1ab2ac29Aa","doi-asserted-by":"crossref","unstructured":"R. H. W.  Hoppe and R.  Kornhuber,\nAdaptive multilevel methods for obstacle problems,\nSIAM J. Numer. Anal. 31 (1994), no. 2, 301\u2013323.","DOI":"10.1137\/0731016"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_030_w2aab3b7d994b1b6b1ab2ac30Aa","doi-asserted-by":"crossref","unstructured":"S.  H\u00fceber and B. I.  Wohlmuth,\nAn optimal a priori error estimate for nonlinear multibody contact problems,\nSIAM J. Numer. Anal. 43 (2005), no. 1, 156\u2013173.","DOI":"10.1137\/S0036142903436678"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_031_w2aab3b7d994b1b6b1ab2ac31Aa","doi-asserted-by":"crossref","unstructured":"S.  Kesavan,\nFunctional Analysis,\nTexts Read. Math. 52,\nHindustan Book Agency, New Delhi, 2009.","DOI":"10.1007\/978-93-86279-42-2"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_032_w2aab3b7d994b1b6b1ab2ac32Aa","doi-asserted-by":"crossref","unstructured":"D.  Kinderlehrer and G.  Stampacchia,\nAn Introduction to Variational Inequalities and Their Applications. Reprint of the 1980 original,\nClass. Appl. Math. 31,\nSociety for Industrial and Applied Mathematics, Philadelphia, 2000.","DOI":"10.1137\/1.9780898719451"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_033_w2aab3b7d994b1b6b1ab2ac33Aa","doi-asserted-by":"crossref","unstructured":"R. H.  Nochetto, K. G.  Siebert and A.  Veeser,\nFully localized a posteriori error estimators and barrier sets for contact problems,\nSIAM J. Numer. Anal. 42 (2005), no. 5, 2118\u20132135.","DOI":"10.1137\/S0036142903424404"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_034_w2aab3b7d994b1b6b1ab2ac34Aa","doi-asserted-by":"crossref","unstructured":"R. H.  Nochetto, T.  von Petersdorff and C.-S.  Zhang,\nA posteriori error analysis for a class of integral equations and variational inequalities,\nNumer. Math. 116 (2010), no. 3, 519\u2013552.","DOI":"10.1007\/s00211-010-0310-y"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_035_w2aab3b7d994b1b6b1ab2ac35Aa","doi-asserted-by":"crossref","unstructured":"M.  Page and D.  Praetorius,\nConvergence of adaptive FEM for some elliptic obstacle problem,\nAppl. Anal. 92 (2013), no. 3, 595\u2013615.","DOI":"10.1080\/00036811.2011.631916"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_036_w2aab3b7d994b1b6b1ab2ac36Aa","doi-asserted-by":"crossref","unstructured":"A.  Schr\u00f6der,\nMixed finite element methods of higher-order for model contact problems,\nSIAM J. Numer. Anal. 49 (2011), no. 6, 2323\u20132339.","DOI":"10.1137\/090770072"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_037_w2aab3b7d994b1b6b1ab2ac37Aa","doi-asserted-by":"crossref","unstructured":"L. R.  Scott and S.  Zhang,\nFinite element interpolation of nonsmooth functions satisfying boundary conditions,\nMath. Comp. 54 (1990), no. 190, 483\u2013493.","DOI":"10.1090\/S0025-5718-1990-1011446-7"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_038_w2aab3b7d994b1b6b1ab2ac38Aa","doi-asserted-by":"crossref","unstructured":"K. G.  Siebert and A.  Veeser,\nA unilaterally constrained quadratic minimization with adaptive finite elements,\nSIAM J. Optim. 18 (2007), no. 1, 260\u2013289.","DOI":"10.1137\/05064597X"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_039_w2aab3b7d994b1b6b1ab2ac39Aa","unstructured":"F. T.  Suttmeier,\nNumerical Solution of Variational Inequalities by Adaptive Finite Elements,\nVieweg+Teubner, Wiesbaden, 2008."},{"key":"2023033109580933521_j_cmam-2017-0018_ref_040_w2aab3b7d994b1b6b1ab2ac40Aa","doi-asserted-by":"crossref","unstructured":"A.  Veeser,\nEfficient and reliable a posteriori error estimators for elliptic obstacle problems,\nSIAM J. Numer. Anal. 39 (2001), no. 1, 146\u2013167.","DOI":"10.1137\/S0036142900370812"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_041_w2aab3b7d994b1b6b1ab2ac41Aa","doi-asserted-by":"crossref","unstructured":"F.  Wang, W.  Han and X.-L.  Cheng,\nDiscontinuous Galerkin methods for solving elliptic variational inequalities,\nSIAM J. Numer. Anal. 48 (2010), no. 2, 708\u2013733.","DOI":"10.1137\/09075891X"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_042_w2aab3b7d994b1b6b1ab2ac42Aa","doi-asserted-by":"crossref","unstructured":"F.  Wang, W.  Han, J.  Eichholz and X.  Cheng,\nA posteriori error estimates for discontinuous Galerkin methods of obstacle problems,\nNonlinear Anal. Real World Appl. 22 (2015), 664\u2013679.","DOI":"10.1016\/j.nonrwa.2014.08.011"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_043_w2aab3b7d994b1b6b1ab2ac43Aa","doi-asserted-by":"crossref","unstructured":"L.  Wang,\nOn the quadratic finite element approximation to the obstacle problem,\nNumer. Math. 92 (2002), no. 4, 771\u2013778.","DOI":"10.1007\/s002110100368"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_044_w2aab3b7d994b1b6b1ab2ac44Aa","doi-asserted-by":"crossref","unstructured":"A.  Weiss and B. I.  Wohlmuth,\nA posteriori error estimator and error control for contact problems,\nMath. Comp. 78 (2009), no. 267, 1237\u20131267.","DOI":"10.1090\/S0025-5718-09-02235-2"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_045_w2aab3b7d994b1b6b1ab2ac45Aa","doi-asserted-by":"crossref","unstructured":"A.  Weiss and B. I.  Wohlmuth,\nA posteriori error estimator for obstacle problems,\nSIAM J. Sci. Comput. 32 (2010), no. 5, 2627\u20132658.","DOI":"10.1137\/090773921"},{"key":"2023033109580933521_j_cmam-2017-0018_ref_046_w2aab3b7d994b1b6b1ab2ac46Aa","doi-asserted-by":"crossref","unstructured":"Q.  Zou, A.  Veeser, R.  Kornhuber and C.  Gr\u00e4ser,\nHierarchical error estimates for the energy functional in obstacle problems,\nNumer. Math. 117 (2011), no. 4, 653\u2013677.","DOI":"10.1007\/s00211-011-0364-5"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/18\/2\/article-p223.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2017-0018\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2017-0018\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T10:33:46Z","timestamp":1680258826000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2017-0018\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,7,11]]},"references-count":46,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2017,6,7]]},"published-print":{"date-parts":[[2018,4,1]]}},"alternative-id":["10.1515\/cmam-2017-0018"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2017-0018","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"value":"1609-9389","type":"electronic"},{"value":"1609-4840","type":"print"}],"subject":[],"published":{"date-parts":[[2017,7,11]]}}}